Abstract
We show that the Mann and Ishikawa iterations are equivalently used to approximate fixed points of generalized contractions.
Authors
S.M. Soltuz
Tiberiu Popoviciu Institute of Numerical Analysis
B.E. Rhoades
Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
Keywords
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Paper coordinates
B.E. Rhoades, Ş.M. Şoltuz, The equivalence between the Mann and Ishikawa iterations dealing with generalized contractions, Int. J. Math. Math. Sci. 2006 (2006), no. 6, 54653.
About this paper
Journal
International Journal of Mathematics and Mathematical Sciences
Publisher Name
Wiley
DOI
Print ISSN
0161-1712
Online ISSN
1687-0425
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[1] S. Ishikawa, Fixed points by a new iteration method, Proceedings of the American MathematicalSociety 44 (1974), no. 1, 147–150.
[2] W. R. Mann, Mean value methods in iteration, Proceedings of the American Mathematical Soci-ety 4 (1953), no. 3, 506–510.
[3] B. E. Rhoades, Convergence of an Ishikawa-type iteration scheme for a generalized contraction, Journal of Mathematical Analysis and Applications 185 (1994), no. 2, 350–355.
[4] B. E. Rhoades and S¸ . M. Soltuz, On the equivalence of Mann and Ishikawa iteration methods,International Journal of Mathematics and Mathematical Sciences 2003 (2003), no. 7, 451–459.
[5] , _____The equivalence between the convergences of Ishikawa and Mann iterations for an asymp-totically pseudocontractive map, Journal of Mathematical Analysis and Applications 283 (2003),no. 2, 681–688.
[6] ,______ The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators,International Journal of Mathematics and Mathematical Sciences 2003 (2003), no. 42, 2645–2651.
[7] , ______The equivalence between the convergences of Ishikawa and Mann iterations for an asymp-totically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps,Journal of Mathematical Analysis and Applications 289 (2004), no. 1, 266–278.
[8] , ______The equivalence of Mann iteration and Ishikawa iteration for a Lipschitzian ψ-uniformlypseudocontractive and ψ-uniformly accretive maps, Tamkang Journal of Mathematics 35 (2004),no. 3, 235–245.
[9] S. M. Soltuz, The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contract-ive operators, Mathematical Communications 10 (2005), no. 1, 81–88.
[10] X. Weng, Fixed point iteration for local strictly pseudo-contractive mapping, Proceedings of the American Mathematical Society 113 (1991), no. 3, 727–731
Paper (preprint) in HTML form
THE EQUIVALENCE BETWEEN THE MANN AND ISHIKAWA ITERATIONS DEALING WITH GENERALIZED CONTRACTIONS
Received 25 January 2006; Revised 10 May 2006; Accepted 11 May 2006
We show that the Mann and Ishikawa iterations are equivalently used to approximate fixed points of generalized contractions.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Let XX be a real Banach space, AA a nonempty convex subset of X,TX, T a selfmap of AA, and let x_(0)=u_(0)in Ax_{0}=u_{0} \in A. The Mann iteration (see [2]) is defined by
{:(1.1)u_(n+1)=(1-alpha_(n))u_(n)+alpha_(n)Tu_(n):}\begin{equation*}
u_{n+1}=\left(1-\alpha_{n}\right) u_{n}+\alpha_{n} T u_{n} \tag{1.1}
\end{equation*}
The Ishikawa iteration is defined (see [1]) by
{:[x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)Ty_(n)],[(1.2)y_(n)=(1-beta_(n))x_(n)+beta_(n)Tx_(n)]:}\begin{gather*}
x_{n+1}=\left(1-\alpha_{n}\right) x_{n}+\alpha_{n} T y_{n} \\
y_{n}=\left(1-\beta_{n}\right) x_{n}+\beta_{n} T x_{n} \tag{1.2}
\end{gather*}
where {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1).
These methods were applied, in [3], to a class of functions TT satisfying the inequality
These methods were applied, in [3], to a class of functions TT satisfying the inequality
{:(1.3)||Tx-Ty|| <= Q(M(x","y)):}\begin{equation*}
\|T x-T y\| \leq Q(M(x, y)) \tag{1.3}
\end{equation*}
where QQ is a real-valued function satisfying
(a) 0 < Q(s) < s0<Q(s)<s for each s > 0s>0 and Q(0)=0Q(0)=0,
(b) QQ is nondecreasing on (0,oo)(0, \infty),
(c) g(s):=s//(s-Q(s))g(s):=s /(s-Q(s)) is nonincreasing on ( 0,oo0, \infty ),
(a) 0 < Q(s) < s0<Q(s)<s for each s > 0s>0 and Q(0)=0Q(0)=0,
(b) QQ is nondecreasing on (0,oo)(0, \infty),
(c) g(s):=s//(s-Q(s))g(s):=s /(s-Q(s)) is nonincreasing on ( 0,oo0, \infty ),
{:(1.4)M(x","y):=max{||x-y||","||x-Tx||","||y-Ty||","||x-Ty||","||y-Tx||}.:}\begin{equation*}
M(x, y):=\max \{\|x-y\|,\|x-T x\|,\|y-T y\|,\|x-T y\|,\|y-T x\|\} . \tag{1.4}
\end{equation*}
In [4], the following conjecture was given: "if the Mann iteration converges, then so does the Ishikawa iteration." In a series of papers [4-8], the authors have given a positive answer to this conjecture, showing, for example, the equivalence between Mann and
Ishikawa iterations for strongly and uniformly pseudocontractive maps. In this note, we show that the convergence of Mann iteration is equivalent to the convergence of Ishikawa iteration, used to approximate fixed points of a map which satisfies condition (1.3). Such a map is independent of the class of strongly pseudocontractive maps. The class of generalized contractions satisfying (1.3) generalizes the class of quasi-contractions, see, for example, [9]. Thus, our result generalizes the main [9, Theorem 1], which states the equivalence between Mann and Ishikawa iterations when applied to quasi-contractions.
Lemma 1.1 [10]. Let {a_(n)}\left\{a_{n}\right\} be a nonnegative sequence which satisfies the following inequality:
{:(1.5)a_(n+1) <= (1-lambda_(n))a_(n)+sigma_(n)",":}\begin{equation*}
a_{n+1} \leq\left(1-\lambda_{n}\right) a_{n}+\sigma_{n}, \tag{1.5}
\end{equation*}
where lambda_(n)in(0,1)\lambda_{n} \in(0,1) for all n >= n_(0),sum_(n=1)^(oo)lambda_(n)=oon \geq n_{0}, \sum_{n=1}^{\infty} \lambda_{n}=\infty, and sigma_(n)=o(lambda_(n))\sigma_{n}=o\left(\lambda_{n}\right). Then lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0.
The following result is a lemma from [3, page 351].
Lemma 1.2 [3]. Let AA be a nonempty closed convex subset of a Banach space XX, and TT a self-map of AA satisfying (1.3). Let {alpha_(n)}\left\{\alpha_{n}\right\} satisfy the conditions alpha_(n) > 0\alpha_{n}>0 for all n >= 0n \geq 0 and sum_(n=0)^(oo)alpha_(n)=oo\sum_{n=0}^{\infty} \alpha_{n}=\infty. Then the sequences {x_(n)},{y_(n)},{u_(n)},{Tx_(n)},{Ty_(n)}\left\{x_{n}\right\},\left\{y_{n}\right\},\left\{u_{n}\right\},\left\{T x_{n}\right\},\left\{T y_{n}\right\}, and {Tu_(n)}\left\{T u_{n}\right\} are bounded.
The following result is a lemma from [3, page 351].
Lemma 1.2 [3]. Let AA be a nonempty closed convex subset of a Banach space XX, and TT a self-map of AA satisfying (1.3). Let {alpha_(n)}\left\{\alpha_{n}\right\} satisfy the conditions alpha_(n) > 0\alpha_{n}>0 for all n >= 0n \geq 0 and sum_(n=0)^(oo)alpha_(n)=oo\sum_{n=0}^{\infty} \alpha_{n}=\infty. Then the sequences {x_(n)},{y_(n)},{u_(n)},{Tx_(n)},{Ty_(n)}\left\{x_{n}\right\},\left\{y_{n}\right\},\left\{u_{n}\right\},\left\{T x_{n}\right\},\left\{T y_{n}\right\}, and {Tu_(n)}\left\{T u_{n}\right\} are bounded.
2. Main result
Theorem 2.1. Let AA be a nonempty closed convex subset of a Banach space XX, and TT a self-map of AA satisfying (1.3). Let {alpha_(n)}\left\{\alpha_{n}\right\} satisfy the conditions alpha_(n) > 0\alpha_{n}>0 for all n >= 0n \geq 0 and sum_(n=0)^(oo)alpha_(n)=oo\sum_{n=0}^{\infty} \alpha_{n}=\infty. Denote by x^(**)x^{*} the unique fixed point of TT. Then for u_(0)=x_(0)in Au_{0}=x_{0} \in A, the following are equivalent:
(i) the Mann iteration (1.1) converges to x^(**)x^{*};
(ii) the Ishikawa iteration (1.2) converges to x^(**)x^{*}.
(i) the Mann iteration (1.1) converges to x^(**)x^{*};
(ii) the Ishikawa iteration (1.2) converges to x^(**)x^{*}.
Proof. Lemma 1.2 assures that both Mann and Ishikawa iterations are bounded and hence, in order to prove the equivalence between (1.1) and (1.2), we need to prove that
{:(2.1)lim_(n rarr oo)||x_(n)-u_(n)||=0.:}\begin{equation*}
\lim _{n \rightarrow \infty}\left\|x_{n}-u_{n}\right\|=0 . \tag{2.1}
\end{equation*}
Set
{:[r_(n)=max{s u p(||x_(n)-Ty_(j)||:j >= n)uu s u p(||u_(n)-Tu_(j)||:j >= n):}],[(2.2){: uu s u p(||x_(n)-Tu_(j)||:j >= n)uu s u p(||u_(n)-Ty_(j)||:j >= n)}.]:}\begin{align*}
r_{n}=\max & \left\{\sup \left(\left\|x_{n}-T y_{j}\right\|: j \geq n\right) \cup \sup \left(\left\|u_{n}-T u_{j}\right\|: j \geq n\right)\right. \\
& \left.\cup \sup \left(\left\|x_{n}-T u_{j}\right\|: j \geq n\right) \cup \sup \left(\left\|u_{n}-T y_{j}\right\|: j \geq n\right)\right\} . \tag{2.2}
\end{align*}
Then the following are true:
{:[||x_(n)-Ty_(j)|| <= (1-alpha_(n-1))||x_(n-1)-Ty_(j)||+alpha_(n-1)||Ty_(n-1)-Ty_(j)||],[ <= (1-alpha_(n-1))r_(n-1)+alpha_(n-1)Q(M(y_(n-1),y_(j)))],[ <= (1-alpha_(n-1))r_(n-1)+alpha_(n-1)Q(r_(n-1))","],[(2.3)||u_(n)-Tu_(j)|| <= (1-alpha_(n-1))||u_(n-1)-Tu_(j)||+alpha_(n-1)||Tu_(n-1)-Tu_(j)||],[ <= (1-alpha_(n-1))r_(n-1)+alpha_(n-1)Q(M(u_(n-1),u_(j)))],[ <= (1-alpha_(n-1))r_(n-1)+alpha_(n-1)Q(r_(n-1))","]:}\begin{align*}
\left\|x_{n}-T y_{j}\right\| & \leq\left(1-\alpha_{n-1}\right)\left\|x_{n-1}-T y_{j}\right\|+\alpha_{n-1}\left\|T y_{n-1}-T y_{j}\right\| \\
& \leq\left(1-\alpha_{n-1}\right) r_{n-1}+\alpha_{n-1} Q\left(M\left(y_{n-1}, y_{j}\right)\right) \\
& \leq\left(1-\alpha_{n-1}\right) r_{n-1}+\alpha_{n-1} Q\left(r_{n-1}\right), \\
\left\|u_{n}-T u_{j}\right\| & \leq\left(1-\alpha_{n-1}\right)\left\|u_{n-1}-T u_{j}\right\|+\alpha_{n-1}\left\|T u_{n-1}-T u_{j}\right\| \tag{2.3}\\
& \leq\left(1-\alpha_{n-1}\right) r_{n-1}+\alpha_{n-1} Q\left(M\left(u_{n-1}, u_{j}\right)\right) \\
& \leq\left(1-\alpha_{n-1}\right) r_{n-1}+\alpha_{n-1} Q\left(r_{n-1}\right),
\end{align*}
moreover,
{:[||x_(n)-Tu_(j)|| <= (1-alpha_(n-1))||x_(n-1)-Tu_(j)||+alpha_(n-1)||Ty_(n-1)-Tu_(j)||],[(2.4) <= (1-alpha_(n-1))r_(n-1)+alpha_(n-1)Q(M(y_(n-1),u_(j)))],[ <= (1-alpha_(n-1))r_(n-1)+alpha_(n-1)Q(r_(n-1))","]:}\begin{align*}
\left\|x_{n}-T u_{j}\right\| & \leq\left(1-\alpha_{n-1}\right)\left\|x_{n-1}-T u_{j}\right\|+\alpha_{n-1}\left\|T y_{n-1}-T u_{j}\right\| \\
& \leq\left(1-\alpha_{n-1}\right) r_{n-1}+\alpha_{n-1} Q\left(M\left(y_{n-1}, u_{j}\right)\right) \tag{2.4}\\
& \leq\left(1-\alpha_{n-1}\right) r_{n-1}+\alpha_{n-1} Q\left(r_{n-1}\right),
\end{align*}
also,
{:[||u_(n)-Ty_(j)|| <= (1-alpha_(n-1))||u_(n-1)-Ty_(j)||+alpha_(n-1)||Tu_(n-1)-Ty_(j)||],[(2.5) <= (1-alpha_(n-1))r_(n-1)+alpha_(n-1)Q(M(u_(n-1),y_(j)))],[ <= (1-alpha_(n-1))r_(n-1)+alpha_(n-1)Q(r_(n-1))]:}\begin{align*}
\left\|u_{n}-T y_{j}\right\| & \leq\left(1-\alpha_{n-1}\right)\left\|u_{n-1}-T y_{j}\right\|+\alpha_{n-1}\left\|T u_{n-1}-T y_{j}\right\| \\
& \leq\left(1-\alpha_{n-1}\right) r_{n-1}+\alpha_{n-1} Q\left(M\left(u_{n-1}, y_{j}\right)\right) \tag{2.5}\\
& \leq\left(1-\alpha_{n-1}\right) r_{n-1}+\alpha_{n-1} Q\left(r_{n-1}\right)
\end{align*}
Eventually, one gets the following evaluation:
{:(2.6)r_(n) <= (1-alpha_(n-1))r_(n-1)+alpha_(n-1)Q(r_(n-1))Longleftrightarrowalpha_(n-1)g(r_(n-1)) <= r_(n-1)-r_(n)",":}\begin{equation*}
r_{n} \leq\left(1-\alpha_{n-1}\right) r_{n-1}+\alpha_{n-1} Q\left(r_{n-1}\right) \Longleftrightarrow \alpha_{n-1} g\left(r_{n-1}\right) \leq r_{n-1}-r_{n}, \tag{2.6}
\end{equation*}
which implies that {r_(n)}\left\{r_{n}\right\} is nonincreasing in nn and positive. Hence, there exists lim_(n rarr oo)r_(n)\lim _{n \rightarrow \infty} r_{n}, denoted by r >= 0r \geq 0. Suppose r > 0r>0. From (2.6), we obtain
{:(2.7)alpha_(n-1)g(r) <= alpha_(n-1)g(r_(n-1)) <= r_(n-1)-r_(n)Longleftrightarrow g(r)sum_(k=0)^(n)alpha_(k) <= sum_(k=0)^(n)(r_(k)-r_(k-1))=r_(0)-r_(n+1).:}\begin{equation*}
\alpha_{n-1} g(r) \leq \alpha_{n-1} g\left(r_{n-1}\right) \leq r_{n-1}-r_{n} \Longleftrightarrow g(r) \sum_{k=0}^{n} \alpha_{k} \leq \sum_{k=0}^{n}\left(r_{k}-r_{k-1}\right)=r_{0}-r_{n+1} . \tag{2.7}
\end{equation*}
The right-hand side is bounded and the left-hand side is unbounded. Thus, r=0r=0. Hence,
{:(2.8){:[lim_(n rarr oo)||x_(n)-Tu_(n)||=0",",lim_(n rarr oo)||u_(n)-Ty_(n)||=0],[lim_(n rarr oo)||x_(n)-Ty_(n)||=0",",lim_(n rarr oo)||u_(n)-Tu_(n)||=0]:}:}\begin{array}{ll}
\lim _{n \rightarrow \infty}\left\|x_{n}-T u_{n}\right\|=0, & \lim _{n \rightarrow \infty}\left\|u_{n}-T y_{n}\right\|=0 \\
\lim _{n \rightarrow \infty}\left\|x_{n}-T y_{n}\right\|=0, & \lim _{n \rightarrow \infty}\left\|u_{n}-T u_{n}\right\|=0 \tag{2.8}
\end{array}
Suppose now that the Mann iteration converges, then one has
{:[||x_(n+1)-u_(n+1)|| <= (1-alpha_(n))||x_(n)-u_(n)||+alpha_(n)||Ty_(n)-Tu_(n)||],[(2.9) <= (1-alpha_(n))||x_(n)-u_(n)||+alpha_(n)(||Ty_(n)-x_(n)||+||x_(n)-Tu_(n)||).]:}\begin{align*}
\left\|x_{n+1}-u_{n+1}\right\| & \leq\left(1-\alpha_{n}\right)\left\|x_{n}-u_{n}\right\|+\alpha_{n}\left\|T y_{n}-T u_{n}\right\| \\
& \leq\left(1-\alpha_{n}\right)\left\|x_{n}-u_{n}\right\|+\alpha_{n}\left(\left\|T y_{n}-x_{n}\right\|+\left\|x_{n}-T u_{n}\right\|\right) . \tag{2.9}
\end{align*}
Using (2.8) and (2.9) and Lemma 1.1, with
{:[lambda_(n):=||x_(n)-u_(n)||","],[(2.10)sigma_(n):=alpha_(n)(||Ty_(n)-x_(n)||+||x_(n)-Tu_(n)||)","],[sigma_(n)=o(alpha_(n))","]:}\begin{gather*}
\lambda_{n}:=\left\|x_{n}-u_{n}\right\|, \\
\sigma_{n}:=\alpha_{n}\left(\left\|T y_{n}-x_{n}\right\|+\left\|x_{n}-T u_{n}\right\|\right), \tag{2.10}\\
\sigma_{n}=o\left(\alpha_{n}\right),
\end{gather*}
4 Equivalent iterations for generalized contractions
we have lim_(n rarr oo)lambda_(n)=0\lim _{n \rightarrow \infty} \lambda_{n}=0, that is, (2.1) holds. The relation
{:(2.11)||x_(n)-x^(**)|| <= ||x_(n)-u_(n)||+||x^(**)-u_(n)||rarr0:}\begin{equation*}
\left\|x_{n}-x^{*}\right\| \leq\left\|x_{n}-u_{n}\right\|+\left\|x^{*}-u_{n}\right\| \rightarrow 0 \tag{2.11}
\end{equation*}
leads to the conclusion that Ishikawa iteration converges too. Suppose now that the Ishikawa iteration converges, then one has
{:[||x_(n+1)-u_(n+1)|| <= (1-alpha_(n))||x_(n)-u_(n)||+alpha_(n)||Ty_(n)-Tu_(n)||],[(2.12) <= (1-alpha_(n))||x_(n)-u_(n)||+alpha_(n)(||Ty_(n)-u_(n)||+||u_(n)-Tu_(n)||)]:}\begin{align*}
\left\|x_{n+1}-u_{n+1}\right\| & \leq\left(1-\alpha_{n}\right)\left\|x_{n}-u_{n}\right\|+\alpha_{n}\left\|T y_{n}-T u_{n}\right\| \\
& \leq\left(1-\alpha_{n}\right)\left\|x_{n}-u_{n}\right\|+\alpha_{n}\left(\left\|T y_{n}-u_{n}\right\|+\left\|u_{n}-T u_{n}\right\|\right) \tag{2.12}
\end{align*}
Using (2.8) and (2.12) and Lemma 1.1, with
{:[lambda_(n):=||x_(n)-u_(n)||],[(2.13)sigma_(n):=alpha_(n)(||Ty_(n)-u_(n)||+||u_(n)-Tu_(n)||)],[sigma_(n)=o(alpha_(n))]:}\begin{gather*}
\lambda_{n}:=\left\|x_{n}-u_{n}\right\| \\
\sigma_{n}:=\alpha_{n}\left(\left\|T y_{n}-u_{n}\right\|+\left\|u_{n}-T u_{n}\right\|\right) \tag{2.13}\\
\sigma_{n}=o\left(\alpha_{n}\right)
\end{gather*}
we have lim_(n rarr oo)lambda_(n)=0\lim _{n \rightarrow \infty} \lambda_{n}=0, that is, (2.1) holds. The relation
{:(2.14)||u_(n)-x^(**)|| <= ||x_(n)-u_(n)||+||x_(n)-x^(**)||rarr0:}\begin{equation*}
\left\|u_{n}-x^{*}\right\| \leq\left\|x_{n}-u_{n}\right\|+\left\|x_{n}-x^{*}\right\| \rightarrow 0 \tag{2.14}
\end{equation*}
leads to the conclusion that Mann iteration converges too.
Acknowledgment
The authors are indebted to the referees for carefully reading this note and for making useful suggestions.
References
[1] S. Ishikawa, Fixed points by a new iteration method, Proceedings of the American Mathematical Society 44 (1974), no. 1, 147-150.
[2] W. R. Mann, Mean value methods in iteration, Proceedings of the American Mathematical Society 4 (1953), no. 3, 506-510.
[3] B. E. Rhoades, Convergence of an Ishikawa-type iteration scheme for a generalized contraction, Journal of Mathematical Analysis and Applications 185 (1994), no. 2, 350-355.
[4] B. E. Rhoades and Ş. M. Şoltuz, On the equivalence of Mann and Ishikawa iteration methods, International Journal of Mathematics and Mathematical Sciences 2003 (2003), no. 7, 451-459.
[5] - , The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map, Journal of Mathematical Analysis and Applications 283 (2003), no. 2, 681-688.
[6] _, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, International Journal of Mathematics and Mathematical Sciences 2003 (2003), no. 42, 26452651.
[7] - , The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, Journal of Mathematical Analysis and Applications 289 (2004), no. 1, 266-278.
[8] _, The equivalence of Mann iteration and Ishikawa iteration for a Lipschitzian Psi\Psi-uniformly pseudocontractive and psi\psi-uniformly accretive maps, Tamkang Journal of Mathematics 35 (2004), no. 3, 235-245.
[9] Ş. M. Şoltuz, The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators, Mathematical Communications 10 (2005), no. 1, 81-88.
[10] X. Weng, Fixed point iteration for local strictly pseudo-contractive mapping, Proceedings of the American Mathematical Society 113 (1991), no. 3, 727-731.
B. E. Rhoades: Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
[2] W. R. Mann, Mean value methods in iteration, Proceedings of the American Mathematical Society 4 (1953), no. 3, 506-510.
[3] B. E. Rhoades, Convergence of an Ishikawa-type iteration scheme for a generalized contraction, Journal of Mathematical Analysis and Applications 185 (1994), no. 2, 350-355.
[4] B. E. Rhoades and Ş. M. Şoltuz, On the equivalence of Mann and Ishikawa iteration methods, International Journal of Mathematics and Mathematical Sciences 2003 (2003), no. 7, 451-459.
[5] - , The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map, Journal of Mathematical Analysis and Applications 283 (2003), no. 2, 681-688.
[6] _, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, International Journal of Mathematics and Mathematical Sciences 2003 (2003), no. 42, 26452651.
[7] - , The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, Journal of Mathematical Analysis and Applications 289 (2004), no. 1, 266-278.
[8] _, The equivalence of Mann iteration and Ishikawa iteration for a Lipschitzian Psi\Psi-uniformly pseudocontractive and psi\psi-uniformly accretive maps, Tamkang Journal of Mathematics 35 (2004), no. 3, 235-245.
[9] Ş. M. Şoltuz, The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators, Mathematical Communications 10 (2005), no. 1, 81-88.
[10] X. Weng, Fixed point iteration for local strictly pseudo-contractive mapping, Proceedings of the American Mathematical Society 113 (1991), no. 3, 727-731.
B. E. Rhoades: Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
E-mail address: rhoades@indiana.edu
Ştefan M. Şoltuz: "Tiberiu Popoviciu" Institute of Numerical Analysis, Romanian Academy, Tiberiu Popoviciu, P.O. Box 68-1, 400110 Cluj-Napoca, Romania
E-mail addresses: smsoltuz@gmail.com; soltuzul@yahoo.com
Ştefan M. Şoltuz: "Tiberiu Popoviciu" Institute of Numerical Analysis, Romanian Academy, Tiberiu Popoviciu, P.O. Box 68-1, 400110 Cluj-Napoca, Romania
E-mail addresses: smsoltuz@gmail.com; soltuzul@yahoo.com